Chapter1-Review

1.1TheLanguageofAlgebraPage3–There’sachartwithlanguagethatmightbehelpfulEx:TranslateeachtoaMathematicalModel(Note:Besuretodeclarevariables)1)Onehalfoftheprice2)Thesalepriceisonehalfoftheprice.1.2TheRealNumbersVennDiagramsEx:a)WhatdoesthisVennDiagramtellus?

b)Wherewould“beardeddragons”appear?Howaboutroses?

Let p= price÷ .p=±p }d;yIbiEor 0.5p

Let p= prices = Sale

S = I • pprice

SetsofNumbersNaturalNumbers:N={1,2,3…}WholeNumbers:W={0,1,2,3…}Integers:Z={…-3,-2,-1,0,1,2,3…}={0,±1,±2,±3…}RationalNumbers:Q=Thesetofnumbersthatcanbewrittenasafraction,orthe

setofallnumbersthatareeitherterminatingorrepeating.

Terminating:

4=4.0 − "

#$=-0.25

Repeating: %

&=0.11111111…= '

%%=0.63636363…=

Buthowdowewritethesetinsetnotation?

Q=SquirrelsandPizzas

fbrayes

:W

z TphfiIs 1

Decimal

:6J÷= 0.333 . . .

= 0.J

.

Q= { ÷ , 'I¥ ,. . ? }

{ Fla ,be Z

,bt 0 }

bookings as,

ftp.T#sandbbezmt

Set - builder notation

for the set of Rational Numbers( i.e. fractions )

the÷÷÷÷i

±: ÷÷. ÷

Squirrels is.

Pizzas

Pizzas

sys= ÷ = I each squirrel

gets I pitza

Is = f- ⇒Each squirrel getsat pizza

*¥=¥ 9¥11 ⇒ Fins

Each squirrel gets 0

0 pizza

4- = 0

¥ = § = ?0 squirrels4 pizzas

we don't know

the answer

±o

undefined

÷2 why ? 6÷ 3=2

6=3.2

¥=0 why ? 0:-3 . -0.

0=3.0

¥ =? 3.0=0

3=0 . ?D

How do you remember it ?

¥ = 0 ¥ undefined

&E¥¥=)\→- 4g death

¥=NoPah%§g.i&m0Et

IrrationalNumbers:H=Thesetofallnonterminating,nonrepeatingdecimals.Examples:(Notethatsomesquarerootsarenotirrational.Forexample, )Reals:R=

D= 1.4 FEZ Notlrratiap= 1.414213562 . ...

I:3:Yn.ms?tatkaq#g/lHUQ~

Union

*lrrtakonals Trationals

R

IN W zQ\

• µ. 0

.-1

Ex : fis an element of

4 EN MEH4 EIW HER4 EZ

4 EQ 4=¥=E t.6.lt4 €IH REIN4ER

NEWa Ezof E Qf < R

Ex:There’sonemistakeinthislistofPrimeNumbers.Findit:PrimeNumbers:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,51,53,59,61,67,71,73,79,83,89…CompositeNumbers:4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,50,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,74,75,76,77,78,80,81,82,84,85,86,88,90,92,93,94,95,96,97,98,99,100…DivisibilityRules:2’s:Evennumber3’s:Whenyouaddthedigitsofthenumber,thesumisdivisibleby3

Ex:57=>5+7=1257=3*29

5’s:Thenumberendsin0or5 Ex:4580=5*9166’s:Thenumberisdivisibleby2and39’s:Whenyouaddthedigitsofthenumber,thesumisdivisibleby9 Ex:81=>8+1=910’s:Thenumberendsin0Pronunciationof-6:NegativesixortheoppositeofsixWhat’stheoppositeof-6?-(-6)=6

¥ 3.17

Ends in 0, 2,4 , 6,8

← II" 57

n

A → so}/ \

goesj 3 . 19

9

1654580T

T - 45¥-5*

-0

AbsoluteValueDefinition:Theabsolutevalueofanumberisitsdistancefrom0onthenumberlineEx:Simplify1)|6|2)|-6|3)-|-6|4)|0|Ex:Canyoudivideby0?Whyorwhynot?

= 6 Eh=6 TT

= - (6)= - 6

:0 The Sharks

will eat

you .

1.3OperationswithRealNumbersBIGNOTE:SquareRootsWhatarethesquarerootsof25? 25=5 “Radical25” - 25=-5 “Theoppositeofradical25”ExponentTerminology:34=“3tothe4thpower” 3isthebase 4istheexponentOnacalculator: 34=OrderofOperations8–4*2=0or8?8–4*2=4*2=8 or =8–8=0OrderofOperations:1)GroupingSymbols

• Parentheses()• Brackets[]• Braces{}

2)Exponents3)Division/Multiplication,WorkLefttoRight4)Subtraction/Addition,WorkLefttoRight

5,-5

3 ¥4 Pa 81

.

3 XO 4 Enter 81

• AbsoluteValue

Ex:Simplify1)8–4*22)81-80(3–4|6–7|2)FractionReview:Ex:1)#+ ∙

+-=

2) − +-#

3)#+ ÷

+-

⇒ =

0=81-80(3-41=112 )

⇐

qfI9qE;a¥¥= 81-80 ( 3-II ) +80=161

= 81-80 (3-41)= 81-80

j¥=¥f= 's÷¥i¥=÷=t⇒t⇒

= +32=1

=3 . 's= 8-

9

1.4SimplifyingAlgebraicExpressionsUsingPropertiesofRealNumbersPropertiesofRealNumbers1)TheCommutativePropertyofAddition a+b=b+a

Doesitworkforsubtraction?2)TheCommutativePropertyofMultiplication ab=ba

Doesitworkfordivision?3)TheAssociativePropertyofAddition (a+b)+c=a+(b+c)4)TheAssociativePropertyofMultiplication (ab)c=a(bc)5)TheDistributiveProperty–VisitingtheFamily a(b+c)=ab+ac

.

E × :

3 + 4 = 4+3

× +2 + × = 2 + × + ×

No

100 - 20 ± 20 - l oo

80 ¥ - 8 0

E × : 3 . 4 = 4 . 3

× . 3 = 3 X

6 ÷ 3 I 3÷ 6

2 ¥ I

4 3 + 5 7 + 3 8 = 4 3 + (57+38)= (43 + 57) + 38= ( 100 ) + 38 = 1 3 8

(3 . 7) . 8 I 3 (7 . 8)( 2 1 ) . 8 3 ( 56 )

1 68 = 1 68 ✓

⇒:( b - c) = ab - ac

earns KEY2Destiny

a(b-c)=ab-acIdentities–Specialnumbersthatdon’tchangethevalueofothernumbers1)AdditiveIdentity:2)MultiplicativeIdentity:Ex:Simplify1)6 0

+ −1# + .

2)-0.5[3(x–4)–2(x+2)]–0.7(x–3)

O Ex : 0+4=43 to =3

I Ex :1 C-2) = . 2

( i 4) 1=14

=¥¥,

-EE,t6x= 2×-3 yt 6x = 8×-3 y

FT

= - 0.5 [ 3×-12-4×+2 )] - 0.7 ( x - 3)

= - 0. 5[x¥2I4]- 0.7 ( x - 3)

= -0.5-[1-16]- 0.7 ( x - 3)

=- 0.5×+0.546 )-0.7¥

= - 0.5×+0.546 ) - 0.7×+0.76 )

= - 0.5×+8 - 0.7×+2 .I

= - Fx+ To.

Fm

1.5SolvingLinearEquationsUsingPropertiesofEqualityWhenyou’resolvinganequation,whatstepsareyouallowedtouse?1)Simplifying 2x–x=62)Addingtobothsides x–4=103)Subtractingfrombothsides(technicallyjustaddingtheopposite) y+3=-64)Multiplyingbothsidesbythesamenumber(alsoknownasclearingfractions) ½x=165)Dividingbothsidesbythesamenumber(again,technicallythisisjustmultiplying) 2x=16

÷.

:

*t#× = 14

-3 -3

y=:zx = 2.16

. 1=32

¥=¥x=8

Ex:Solveandcheck 1)+45+ − 42 = 2 −54## 2)0.02x+0.0175(110–x)=2.175Question:Whatdoyoudowhenanequationdoesn’thaveasolution?

= ante '

terms

3,2 ⇒ LcD=6

265.33¥.61=6.2-¥.t¥t

2(3tp ) - 24p= 12-34>+2 )

6+2p - 2413=12 - 3p - 6

6- ZZP = 6 - 3p-

= -4 y10,0000 .oD

Clear decimals : 10000 = 0. 02004- new

10,000 (0.02A + 10,000 [0.0175410×5]=10,000/2.175)200×+175410- D= 21750

200×+19250

Ex :

.xxt.FI,TO False

No Solution

Equation : 0=1 FalseNo Solution

( Done - Nocheckng)

Single Expression :

f- undefined

FZ Undefined

or Not Real

or Imaginary

6 -ZZP = 6-3p

+221>+26= 6+19 p

- 6 - 6-

0 = 19 p

O_ =19PM19€

Check : 3¥ - 4p= 2 -

¥

3¥ .

4(dE 2- 0¥

=⇒o÷F÷-

✓

Factors : 3p

Terms : 3+p or 3 -

p

Ex : Simplify�1� }I=¥- =p I

MultiplicationIs you

�2� 3¥ =Can't be

Simplifiedbecause additionIs stupid

1.6SolvingFormulas;GeometryFormulasyoushouldknowbyheart:

Area= Perimeter=

Area= Circumference(theperimeterofacircle)=

Area=

Perimeter=

Area=

Ex:Sixty-sixfeetofcrownmoldingwasinstalledaroundarectangulardiningroom.Ifthelengthoftheroomis18.75feet,whatisitswidth?Ex:Findtheareaofacirclewithdiameter3.8ft.

w ,€90°=right anglelw . length . width

Rectangle 2l+2w=2( ltw )Tr2TryA D= Ztrarg?]n tzbjha

,+sibz+sib3base - b

Trianglea

"

;←h÷(a+b)h

b

Trapezoid

the 88IPTwiIdE66=20+248.75 )66.2W +37.5

- 37.5 - 37.5

28¥

t€AE'¥4.95 /"IE¥ff+d=3.8ft

= 3.61 'T ft'-

r=÷( 3. 8) =L-9ft}~~ 11.34ft

'

Edu A=bh§g

h :# A= ah too small

f€fA=fCa+b)hjust right.

SolvingforaSpecifiedVariableEx:SolveA=P+Prtfor a)t b)P

not

.

A=P+Prt⇒t=¥n. pp

APT

A¥=¥¥÷⇒A¥=t⇒⇐n¥

A=P+Prt

A=P( itrt )

i¥='YEH'

ft÷=P

P + A +6=12317

Solve for P

Pta +6 =P -317-

PA+6 = - 3PrA + 6 = Pf3r )

tea

A3¥=P@p= A +6

=- ( At 6)

⇒ a= -As÷

1.7UsingEquationstoSolveProblemsWordProblemstoReviewEx:(#12,Page71)AsofOctober2010,DenzelWashington’sthreetopdomesticgrowingfilms,AmericanGangster,RemembertheTitans,andThePelicanBrief,hadearnedatotalof$346.7million.IfAmericanGangsterearned$14.5millionmorethanRemembertheTitans,andifRemembertheTitansearned$14.9millionmorethanThePelicanBrief,howmuchdideachfilmearnasofthatdate?Ex:(#22,Page72)Astudentworkingforadeliverycompanyearns$57.50perdayplus$4.75foreachpackageshedelivers.Howmanydeliveriesmustshemakeeachdaytoearn$200aday?

s. r = 14.9+17

Let a = American Gangsters.

earnings -

r= Remember the Titan 's " =)p= Thepelican Brief 's "

346.7 -- Total = atrtp

a = r +14.5

T Tbig smaller

Let D= # deliveries per day

Daily wage + Deliveries = Total

57.50+4.752= 200

Fixed -9 Variables

57.50T 4.754=200

-57.50-57.504.751=142.50

D= 142.504.75deliveries or

D= 30 papcdadagfs

at rtp = 346.7

a = rt 14.5

r = 14.9 + p

Substitution : fthrjfriablensa + r + p = 346.7

Gt 14.5 )+ 449 + p ) + p= 346.7

rt Zp + 29.4 = 346.7C- Two variables in

( 14.9 + p ) + Zpt 29.4 = 346.7~

3p + 44.3 = 346.7

- - 44.3 - 44.3One Variable -

y 3p = 302.4

p = 100.8

17=100.8

r= 14.9+ p =14.9+100.8= 115.7

a = 14.5T r = 14.5+115.7= 130.2

Totak 100.8+115.7+130.2

= 346.7 ✓ 0

= $100.8 million%=$115.7 million

a= $130.2 million

Ex:(#36,Page73)Intheillustration,r||s(readas“linerisparalleltolines”),andb=137.Findaandc.1.8MoreaboutProblemSolvingMoreWordProblemstoReview,includingMixtureProblemsEx:(#16,Page84)Completethefollowingtablethatcouldbeusedtosolvethisproblem:Howmanypintsofpunchfromtheorangecoolermustbemixedwiththeentirecontentsofthebluecoolertogeta12%punchmixture?Thenwriteanequationthatcouldbeusedtosolvethisliquidmeasureproblem. Amount*Strength=Pureconcentrate

Strong 20

Weak 0.10

Mix

If÷370 =) a= 1370

137T C = 180

-13=37C = 430

ftp..fm#"

rpints.jo?YzoqYYzrIYf+=4+o.ir=o.iyzotD

2.Is @Engage.

T.ge# m

l 11 m l is parallel to M

L ) +D.= 180°

A 1800

@36.0°Jr 90°